{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:42:16Z","timestamp":1740123736021,"version":"3.37.3"},"reference-count":16,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,6,15]],"date-time":"2021-06-15T00:00:00Z","timestamp":1623715200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,6,15]],"date-time":"2021-06-15T00:00:00Z","timestamp":1623715200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004270","name":"Royal Institute of Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004270","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2022,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In the two parts of the present note we discuss questions concerning the implementation of overdetermined least-squares collocation methods for higher index differential-algebraic equations (DAEs). Since higher index DAEs lead to ill-posed problems in natural settings, the discrete counterparts are expected to be very sensitive, which attaches particular importance to their implementation. We provide in Part 1 a robust selection of basis functions and collocation points to design the discrete problem whereas we analyze the discrete least-squares problem and substantiate a procedure for its numerical solution in Part 2.<\/jats:p>","DOI":"10.1007\/s11075-021-01141-6","type":"journal-article","created":{"date-parts":[[2021,6,15]],"date-time":"2021-06-15T10:03:18Z","timestamp":1623751398000},"page":"965-986","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Towards a reliable implementation of least-squares collocation for higher index differential-algebraic equations\u2014Part 2: the discrete least-squares problem"],"prefix":"10.1007","volume":"89","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4950-6646","authenticated-orcid":false,"given":"Michael","family":"Hanke","sequence":"first","affiliation":[]},{"given":"Roswitha","family":"M\u00e4rz","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,6,15]]},"reference":[{"key":"1141_CR1","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1137\/0908047","volume":"8","author":"U Ascher","year":"1987","unstructured":"Ascher, U., Bader, G.: A new basis implementation for a mixed order boundary-value ode solver. SIAM J. Sci Statist. Comput. 8, 483\u2013500 (1987)","journal-title":"SIAM J. Sci Statist. Comput."},{"key":"1141_CR2","doi-asserted-by":"crossref","unstructured":"Barlow, J.L.: Solution of sparse weighted and equality constrained least squares problems. In: Page, C., LePage, R. (eds.) Computing Science and Statistics, pp 53\u201362. Springer, New York (1992)","DOI":"10.1007\/978-1-4612-2856-1_7"},{"issue":"1","key":"1141_CR3","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1137\/0729016","volume":"29","author":"JL Barlow","year":"1992","unstructured":"Barlow, J.L., Vemulapati, U.B.: A note on deferred correction for equality constrained least squares problems. SIAM J. Numer Anal. 29(1), 249\u2013256 (1992)","journal-title":"SIAM J. Numer Anal."},{"key":"1141_CR4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971484","volume-title":"Numerical Methods for Least Squares Problems","author":"\u00c5 Bj\u00f6rck","year":"1996","unstructured":"Bj\u00f6rck, \u00c5.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)"},{"key":"1141_CR5","doi-asserted-by":"publisher","first-page":"322","DOI":"10.1007\/BF01939326","volume":"7","author":"\u00c5 Bj\u00f6rck","year":"1967","unstructured":"Bj\u00f6rck, \u00c5., Golub, G.H.: Iterative refinement of linear least squares solutions by Householder transformations. BIT 7, 322\u2013337 (1967)","journal-title":"BIT"},{"key":"1141_CR6","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1007\/s002110050099","volume":"69","author":"SL Campbell","year":"1995","unstructured":"Campbell, S.L., Moore, E.: Constraint preserving integrators for general nonlinear higher index DAEs. Num. Math. 69, 383\u2013399 (1995)","journal-title":"Num. Math."},{"key":"1141_CR7","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718881","volume-title":"Direct Methods for Sparse Linear Systems. Fundamentals of Algorithms","author":"TA Davis","year":"2006","unstructured":"Davis, T.A.: Direct Methods for Sparse Linear Systems. Fundamentals of Algorithms. SIAM, Philadelphia (2006)"},{"issue":"1","key":"1141_CR8","first-page":"8:1","volume":"38","author":"TA Davis","year":"2011","unstructured":"Davis, T.A.: Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization. ACM Trans. Math. Softw. 38 (1), 8:1\u20138:22 (2011)","journal-title":"ACM Trans. Math. Softw."},{"key":"1141_CR9","volume-title":"Matrix Computations","author":"GH Golub","year":"1989","unstructured":"Golub, G.H., van Loan, C.h.: Matrix Computations, 2nd edn. The Johns Hopkins University Press, Baltimore and London (1989)","edition":"2nd edn."},{"key":"1141_CR10","unstructured":"Guennebaud, G., Jacob, B., et al.: Eigen v3. http:\/\/eigen.tuxfamily.org(2010)"},{"key":"1141_CR11","unstructured":"Hanke, M., M\u00e4rz, R.: Towards a reliable implementation of least-squares collocation for higher-index linear differential-algebaic equations. Part 1: Basics and ansatz choices. Numerical Algorithms. submitted"},{"issue":"318","key":"1141_CR12","doi-asserted-by":"publisher","first-page":"1647","DOI":"10.1090\/mcom\/3393","volume":"88","author":"M Hanke","year":"2019","unstructured":"Hanke, M., M\u00e4rz, R., Tischendorf, C.: Least-squares collocation for higher-index linear differential-algebaic equations Estimating the stability threshold. Math. Comp. 88(318), 1647\u20131683 (2019). https:\/\/doi.org\/10.1090\/mcom\/3393","journal-title":"Math. Comp."},{"key":"1141_CR13","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1016\/j.cam.2016.12.017","volume":"317","author":"M Hanke","year":"2017","unstructured":"Hanke, M., M\u00e4rz, R., Tischendorf, C., Weinm\u00fcller, E., Wurm, S.: Least-squares collocation for linear higher-index differential-algebraic equations. J. Comput. Appl Math. 317, 403\u2013431 (2017). https:\/\/doi.org\/10.1016\/j.cam.2016.12.017","journal-title":"J. Comput. Appl Math."},{"key":"1141_CR14","doi-asserted-by":"crossref","unstructured":"Lamour, R., M\u00e4rz, R., Tischendorf, C. In: Ilchmann, A., Reis, T. (eds.) : Differential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum. Springer-Verlag, Berlin Heidelberg New York Dordrecht London (2013)","DOI":"10.1007\/978-3-642-27555-5"},{"key":"1141_CR15","unstructured":"Stallman, R.M., GCC Developers Community, et al: Using the Gnu Compiler collection. CreateSpace Scotts Valley (2009)"},{"issue":"5","key":"1141_CR16","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1137\/0722051","volume":"22","author":"Ch van Loan","year":"1985","unstructured":"van Loan, C.h.: On the method of weighting for equality-constrained least-squares problems. SIAM J. Numer. Anal. 22(5), 851\u2013864 (1985)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-021-01141-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-021-01141-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-021-01141-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,2,4]],"date-time":"2022-02-04T17:18:34Z","timestamp":1643995114000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-021-01141-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,15]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,3]]}},"alternative-id":["1141"],"URL":"https:\/\/doi.org\/10.1007\/s11075-021-01141-6","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2021,6,15]]},"assertion":[{"value":"14 May 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 May 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 June 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}